Given a set of fertility rates \(\tilde{f}_i\) on age intervals \([x_{i-1},x_i)\) ,\(i=1,\ldots,N\) calculate the cummulative fertility rate \(F_i\) at \(x_i\) $$ F_i=\left\{ \begin{array}{ll} 0 & ,i=0\\ \sum_{k\le i} (x_k-x_{k-1}) \tilde{f}_k & ,i > 0 \end{array}\right. $$ at \(x_i\). Find fertility \(f_i\) at internal points \(x_i\) where \(i=1,\ldots,N-1\) by linear interpolating \(\tilde{f}\) $$ f_i = \tilde{f}_{i+1}\frac{x_i-x_{i-1}}{x_{i+1}-x_{i-1}} + \tilde{f}_i\frac{x_{i+1}-x_i}{x_{i+1}-x_{i-1}} $$ Find \(f_0\) at \(x_0\) and \(f_N\) at \(x_N\) by assuming fertility is linear in age over the end intervals \([x_0,x_1]\) and \([x_{N-1},x_{N}]\) $$ \begin{eqnarray} f_0&=&2\tilde{f}_1 - f_1\\ f_N&=& 2\tilde{f}_N - f_{N-1} \end{eqnarray} $$ For the internal intervals \([x_{i-1},x_i]\) for \(i=2,\ldots,N-1\) assume fertility is quadratic with curvature \(C_i\) $$ C_i=\frac{12}{(x_i-x_{i-1})^2}\left(\frac{f_i+f_{i-1}}{2}-\tilde{f}_i\right) $$